Median and MVUE

Attention Conservation Notice: optimality in a one-dimensional parametric model is so 1960s. Wikipedia says “For practical statistics problems, it is important to determine the MVUE if one exists, since less-than-optimal procedures would naturally be avoided, other things being equal”, but they’re not using the word “practical” to mean what I would use it to mean In the Laplace (or double exponential) distribution, with density \[f(x;\theta) =\frac{1}{2} \exp(-|x-\theta|)\] the maximum likelihood estimator of \(\theta\) is the median.